I believe you have a '62. Just have the Steering Box re-adjusted.
That's all it may need.
BUT, have the complete Front End checked over first. You may have some worn parts that could be contributing to the play.
I've driven Chevys from '55 to '72 with no problems other than a minor adjustment here and there on the earlier ones. Later ones ('65-'72) never gave me an ounce of trouble through the years. Never even needed an adjustment.
-Parting Out over 75 '58 to '73 Full Size Chevy Cars-
-National Impala Association
-Vintage Chevrolet Club of America